{"id":2995,"date":"2023-05-17T18:41:20","date_gmt":"2023-05-17T18:41:20","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=2995"},"modified":"2025-08-26T03:25:28","modified_gmt":"2025-08-26T03:25:28","slug":"triangles-learn-it-3","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/triangles-learn-it-3\/","title":{"raw":"Triangles: Learn It 3","rendered":"Triangles: Learn It 3"},"content":{"raw":"<h2>Similar Triangles<\/h2>\r\n<p>When we use a map to plan a trip, a sketch to build a bookcase, or a pattern to sew a dress, we are working with similar figures. In geometry, if two figures have exactly the same shape but different sizes, we say they are <strong>similar figures<\/strong>. One is a scale model of the other. The corresponding sides of the two figures have the same ratio, and all their corresponding angles are have the same measures.<\/p>\r\n<p>The two triangles below\u00a0are similar. Each side of [latex]\\Delta ABC[\/latex] is four times the length of the corresponding side of [latex]\\Delta XYZ[\/latex] and their corresponding angles have equal measures.<\/p>\r\n<p>[latex]\\Delta ABC[\/latex] and [latex]\\Delta XYZ[\/latex] are similar triangles. Their corresponding sides have the same ratio and the corresponding angles have the same measure.<\/p>\r\n<center>\r\n[caption id=\"\" align=\"aligncenter\" width=\"522\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223723\/CNX_BMath_Figure_09_03_020.png\" alt=\"Two triangles that appear to be the same shape, but the triangle on the right is smaller. The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled 16, the side across from B is labeled 20, and the side across from C is labeled 12. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 4, the side across from Y is labeled 5, and the side across from Z is labeled 3. Beside the triangles, it says that the measure of angle A equals the measure of angle X, the measure of angle B equals the measure of angle Y, and the measure of angle C equals the measure of angle Z. Below this is the proportion 16 over 4 equals 20 over 5 equals 12 over 3.\" width=\"522\" height=\"140\" \/> Figure 1. Triangle ABC and triangle XYZ are similar figures[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>properties of similar triangles<\/h3>\r\n<p>If two triangles are similar, then their corresponding angle measures are equal and their corresponding side lengths are in the same ratio.<\/p>\r\n<p>&nbsp;<\/p>\r\n<center><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223725\/CNX_BMath_Figure_09_03_056_img-1.png\" alt=\"Two triangles that appear to be the same shape, but the triangle on the right is smaller. The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled a, the side across from B is labeled b, and the side across from C is labeled c. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled x, the side across from Y is labeled y, and the side across from Z is labeled z. The vertices are labeled with capital letters and the sides are labeled with lowercase letters. Beside the triangles, it says that the measure of angle A equals the measure of angle X, the measure of angle B equals the measure of angle Y, and the measure of angle C equals the measure of angle Z. Below this is a proportion of only lowercase letters reading a over x equals b over y equals c over z. \" width=\"628\" height=\"176\" \/><\/center>\r\n<p>&nbsp;<\/p>\r\n<\/div>\r\n<\/section>\r\n<p>The length of a side of a triangle may be referred to by its endpoints, two vertices of the triangle. For example, in [latex]\\Delta ABC\\text{:}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\text{the length }a\\text{ can also be written }BC\\hfill \\\\ \\text{the length }b\\text{ can also be written }AC\\hfill \\\\ \\text{the length }c\\text{ can also be written }AB\\hfill \\end{array}[\/latex]<\/p>\r\n<p>We will often use this notation when we solve similar triangles because it will help us match up the corresponding side lengths.<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]7026[\/ohm2_question]<\/section>\r\n<p>We can use the notion that triangles are similar to solve for an unknown side length.<\/p>\r\n<section class=\"textbox questionHelp\">\r\n<p><strong>How To: Find the Unknown Side Length in a Triangle When Given Similar Triangles<\/strong><\/p>\r\n<p>To find an unknown side length in a triangle when given similar triangles, you can use the concept of proportional relationships between corresponding sides. Below are steps to follow when solving for an unknown side given similar triangles. We will use the following triangles to demonstrate these steps:<\/p>\r\n<p>&nbsp;<\/p>\r\n<center><img class=\"aligncenter wp-image-3005\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/17185658\/shutterstock_2149824621-scaled.jpg\" alt=\"two similar triangles. Triangle ABC has sides labeled AB 18, BC 12, and CA x. Triangle EDF es labeled DE y, EF 9, and FD 18.. Angle A is equal to angle D, angle B is equal to angle E and angle C is equal to angle F.\" width=\"334\" height=\"200\" \/><\/center>\r\n<p>&nbsp;<\/p>\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>Identify the corresponding sides in the two similar triangles.\r\n\r\n<ul>\r\n\t<li>[latex]AB \\sim ED, BC \\sim EF, CA \\sim FD[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>Write the ratio of the lengths of the corresponding sides.\r\n\r\n<ul>\r\n\t<li>[latex]\\frac{AB}{ED} = \\frac{BC}{EF} = \\frac{CA}{FD}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>Set up a proportion to solve for the unknown.\r\n\r\n<ul>\r\n\t<li>[latex]\\frac{AB}{ED} = \\frac{BC}{EF}[\/latex]<\/li>\r\n\t<li>[latex]\\frac{18}{x} = \\frac{12}{9}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>Solve the proportion by cross multiplying and dividing to find the unknown\r\n\r\n<ul>\r\n\t<li>[latex]AB*EF = BC*ED[\/latex]<\/li>\r\n\t<li>[latex]18*9 = 12*x[\/latex]<\/li>\r\n\t<li>[latex]\\frac{18*9}{12} =x[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/section>\r\n<p>Here is an example for you to try.<\/p>\r\n<section class=\"textbox example\">[latex]\\Delta ABC[\/latex] and [latex]\\Delta XYZ[\/latex] are similar triangles. The lengths of two sides of each triangle are shown. Find the lengths of the third side of each triangle.<center><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223727\/CNX_BMath_Figure_09_03_022.png\" alt=\"Two triangles that appear to be the same shape, but the triangle on the right is smaller. The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled a, the side across from B is labeled 3.2, and the side across from C is labeled 4. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 4.5, the side across from Y is labeled y, and the side across from Z is labeled 3.\" width=\"322\" height=\"106\" \/><\/center>\r\n<p>&nbsp;<\/p>\r\n\r\n[reveal-answer q=\"584050\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"584050\"]\r\n\r\n<table id=\"eip-id1168466690156\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\r\n<td>The figure is provided.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\r\n<td>The length of the sides of similar triangles<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\r\n<td>\r\n<p>Let<\/p>\r\n<p>[latex]a[\/latex] = length of the third side of [latex]\\Delta ABC[\/latex]<\/p>\r\n<p>[latex]y[\/latex] = length of the third side [latex]\\Delta XYZ[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Step 4. <strong>Translate.<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">\r\n<p>The triangles are similar, so the corresponding sides are in the same ratio. So<\/p>\r\n<p>[latex]{\\Large\\frac{AB}{XY}}={\\Large\\frac{BC}{YZ}}={\\Large\\frac{AC}{XZ}}[\/latex]<\/p>\r\n<p>Since the side [latex]AB=4[\/latex] corresponds to the side [latex]XY=3[\/latex] , we will use the ratio [latex]{\\Large\\frac{\\mathrm{AB}}{\\mathrm{XY}}}={\\Large\\frac{4}{3}}[\/latex] to find the other sides.<\/p>\r\n<p>Be careful to match up corresponding sides correctly.<\/p>\r\n<p><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223728\/CNX_BMath_Figure_09_03_057_img-01.png\" alt=\"A set of multiple proportions to find a and y. To find a, for the sides of large triangle, the proportion AB over XY equals BC over YZ is shown, and for sides of small triangle, the proportion 4 over 3 is equal to a over 4.5 is shown. To find y, for the sides of large triangle, the proportion AB over XY equals AC over XZ is shown and for the sides of small triangle, the proportion 4 over 3 is equal to 3.2 over y is shown.\" width=\"565\" height=\"122\" \/><\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223730\/CNX_BMath_Figure_09_03_057_img-02.png\" alt=\"Two solved equations. The first shows 3a = 4(4.5), which simplifies to 3a = 18, which simplifies to a = 6. The second shows 4y = 3(3.2), which simplifies to 4y = 9.6, which simplifies to y = 2.4.\" width=\"269\" height=\"78\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p>Step 6. <strong>Check.<\/strong><\/p>\r\n<\/td>\r\n<td>\r\n<p><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223731\/CNX_BMath_Figure_09_03_057_img-03.png\" alt=\"A set of two equations being checked. The first begins with 4 over 3 equals 6 over 4.5, which is changed to 4(4.5) = 6(3), which is changed to 18 = 18, which is true. The second begins with 4 over 3 is equal to 3.2 over 2.4, which is changed to 4(2.4) equals 3.2(3), which is changed to 9.6 = 9.6, which is true.\" width=\"258\" height=\"111\" \/><\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\r\n<td>The third side of [latex]\\Delta ABC[\/latex] is [latex]6[\/latex] and the third side of [latex]\\Delta XYZ[\/latex] is [latex]2.4[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]7027[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]7028[\/ohm2_question]<\/section>","rendered":"<h2>Similar Triangles<\/h2>\n<p>When we use a map to plan a trip, a sketch to build a bookcase, or a pattern to sew a dress, we are working with similar figures. In geometry, if two figures have exactly the same shape but different sizes, we say they are <strong>similar figures<\/strong>. One is a scale model of the other. The corresponding sides of the two figures have the same ratio, and all their corresponding angles are have the same measures.<\/p>\n<p>The two triangles below\u00a0are similar. Each side of [latex]\\Delta ABC[\/latex] is four times the length of the corresponding side of [latex]\\Delta XYZ[\/latex] and their corresponding angles have equal measures.<\/p>\n<p>[latex]\\Delta ABC[\/latex] and [latex]\\Delta XYZ[\/latex] are similar triangles. Their corresponding sides have the same ratio and the corresponding angles have the same measure.<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 522px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223723\/CNX_BMath_Figure_09_03_020.png\" alt=\"Two triangles that appear to be the same shape, but the triangle on the right is smaller. The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled 16, the side across from B is labeled 20, and the side across from C is labeled 12. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 4, the side across from Y is labeled 5, and the side across from Z is labeled 3. Beside the triangles, it says that the measure of angle A equals the measure of angle X, the measure of angle B equals the measure of angle Y, and the measure of angle C equals the measure of angle Z. Below this is the proportion 16 over 4 equals 20 over 5 equals 12 over 3.\" width=\"522\" height=\"140\" \/><figcaption class=\"wp-caption-text\">Figure 1. Triangle ABC and triangle XYZ are similar figures<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>properties of similar triangles<\/h3>\n<p>If two triangles are similar, then their corresponding angle measures are equal and their corresponding side lengths are in the same ratio.<\/p>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223725\/CNX_BMath_Figure_09_03_056_img-1.png\" alt=\"Two triangles that appear to be the same shape, but the triangle on the right is smaller. The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled a, the side across from B is labeled b, and the side across from C is labeled c. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled x, the side across from Y is labeled y, and the side across from Z is labeled z. The vertices are labeled with capital letters and the sides are labeled with lowercase letters. Beside the triangles, it says that the measure of angle A equals the measure of angle X, the measure of angle B equals the measure of angle Y, and the measure of angle C equals the measure of angle Z. Below this is a proportion of only lowercase letters reading a over x equals b over y equals c over z.\" width=\"628\" height=\"176\" \/><\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<\/section>\n<p>The length of a side of a triangle may be referred to by its endpoints, two vertices of the triangle. For example, in [latex]\\Delta ABC\\text{:}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\text{the length }a\\text{ can also be written }BC\\hfill \\\\ \\text{the length }b\\text{ can also be written }AC\\hfill \\\\ \\text{the length }c\\text{ can also be written }AB\\hfill \\end{array}[\/latex]<\/p>\n<p>We will often use this notation when we solve similar triangles because it will help us match up the corresponding side lengths.<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm7026\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=7026&theme=lumen&iframe_resize_id=ohm7026&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<p>We can use the notion that triangles are similar to solve for an unknown side length.<\/p>\n<section class=\"textbox questionHelp\">\n<p><strong>How To: Find the Unknown Side Length in a Triangle When Given Similar Triangles<\/strong><\/p>\n<p>To find an unknown side length in a triangle when given similar triangles, you can use the concept of proportional relationships between corresponding sides. Below are steps to follow when solving for an unknown side given similar triangles. We will use the following triangles to demonstrate these steps:<\/p>\n<p>&nbsp;<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3005\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/17185658\/shutterstock_2149824621-scaled.jpg\" alt=\"two similar triangles. Triangle ABC has sides labeled AB 18, BC 12, and CA x. Triangle EDF es labeled DE y, EF 9, and FD 18.. Angle A is equal to angle D, angle B is equal to angle E and angle C is equal to angle F.\" width=\"334\" height=\"200\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/17185658\/shutterstock_2149824621-scaled.jpg 2560w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/17185658\/shutterstock_2149824621-300x180.jpg 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/17185658\/shutterstock_2149824621-1024x615.jpg 1024w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/17185658\/shutterstock_2149824621-768x461.jpg 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/17185658\/shutterstock_2149824621-1536x922.jpg 1536w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/17185658\/shutterstock_2149824621-2048x1230.jpg 2048w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/17185658\/shutterstock_2149824621-1200x721.jpg 1200w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/17185658\/shutterstock_2149824621-65x39.jpg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/17185658\/shutterstock_2149824621-225x135.jpg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/18\/2023\/05\/17185658\/shutterstock_2149824621-350x210.jpg 350w\" sizes=\"(max-width: 334px) 100vw, 334px\" \/><\/div>\n<p>&nbsp;<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>Identify the corresponding sides in the two similar triangles.\n<ul>\n<li>[latex]AB \\sim ED, BC \\sim EF, CA \\sim FD[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li>Write the ratio of the lengths of the corresponding sides.\n<ul>\n<li>[latex]\\frac{AB}{ED} = \\frac{BC}{EF} = \\frac{CA}{FD}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li>Set up a proportion to solve for the unknown.\n<ul>\n<li>[latex]\\frac{AB}{ED} = \\frac{BC}{EF}[\/latex]<\/li>\n<li>[latex]\\frac{18}{x} = \\frac{12}{9}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li>Solve the proportion by cross multiplying and dividing to find the unknown\n<ul>\n<li>[latex]AB*EF = BC*ED[\/latex]<\/li>\n<li>[latex]18*9 = 12*x[\/latex]<\/li>\n<li>[latex]\\frac{18*9}{12} =x[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/section>\n<p>Here is an example for you to try.<\/p>\n<section class=\"textbox example\">[latex]\\Delta ABC[\/latex] and [latex]\\Delta XYZ[\/latex] are similar triangles. The lengths of two sides of each triangle are shown. Find the lengths of the third side of each triangle.<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223727\/CNX_BMath_Figure_09_03_022.png\" alt=\"Two triangles that appear to be the same shape, but the triangle on the right is smaller. The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled a, the side across from B is labeled 3.2, and the side across from C is labeled 4. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 4.5, the side across from Y is labeled y, and the side across from Z is labeled 3.\" width=\"322\" height=\"106\" \/><\/div>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q584050\">Show Solution<\/button><\/p>\n<div id=\"q584050\" class=\"hidden-answer\" style=\"display: none\">\n<table id=\"eip-id1168466690156\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\n<td>The figure is provided.<\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>The length of the sides of similar triangles<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>\n<p>Let<\/p>\n<p>[latex]a[\/latex] = length of the third side of [latex]\\Delta ABC[\/latex]<\/p>\n<p>[latex]y[\/latex] = length of the third side [latex]\\Delta XYZ[\/latex]<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Step 4. <strong>Translate.<\/strong><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">\n<p>The triangles are similar, so the corresponding sides are in the same ratio. So<\/p>\n<p>[latex]{\\Large\\frac{AB}{XY}}={\\Large\\frac{BC}{YZ}}={\\Large\\frac{AC}{XZ}}[\/latex]<\/p>\n<p>Since the side [latex]AB=4[\/latex] corresponds to the side [latex]XY=3[\/latex] , we will use the ratio [latex]{\\Large\\frac{\\mathrm{AB}}{\\mathrm{XY}}}={\\Large\\frac{4}{3}}[\/latex] to find the other sides.<\/p>\n<p>Be careful to match up corresponding sides correctly.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223728\/CNX_BMath_Figure_09_03_057_img-01.png\" alt=\"A set of multiple proportions to find a and y. To find a, for the sides of large triangle, the proportion AB over XY equals BC over YZ is shown, and for sides of small triangle, the proportion 4 over 3 is equal to a over 4.5 is shown. To find y, for the sides of large triangle, the proportion AB over XY equals AC over XZ is shown and for the sides of small triangle, the proportion 4 over 3 is equal to 3.2 over y is shown.\" width=\"565\" height=\"122\" \/><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223730\/CNX_BMath_Figure_09_03_057_img-02.png\" alt=\"Two solved equations. The first shows 3a = 4(4.5), which simplifies to 3a = 18, which simplifies to a = 6. The second shows 4y = 3(3.2), which simplifies to 4y = 9.6, which simplifies to y = 2.4.\" width=\"269\" height=\"78\" \/><\/td>\n<\/tr>\n<tr>\n<td>\n<p>Step 6. <strong>Check.<\/strong><\/p>\n<\/td>\n<td>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223731\/CNX_BMath_Figure_09_03_057_img-03.png\" alt=\"A set of two equations being checked. The first begins with 4 over 3 equals 6 over 4.5, which is changed to 4(4.5) = 6(3), which is changed to 18 = 18, which is true. The second begins with 4 over 3 is equal to 3.2 over 2.4, which is changed to 4(2.4) equals 3.2(3), which is changed to 9.6 = 9.6, which is true.\" width=\"258\" height=\"111\" \/><\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The third side of [latex]\\Delta ABC[\/latex] is [latex]6[\/latex] and the third side of [latex]\\Delta XYZ[\/latex] is [latex]2.4[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm7027\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=7027&theme=lumen&iframe_resize_id=ohm7027&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm7028\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=7028&theme=lumen&iframe_resize_id=ohm7028&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":15,"menu_order":13,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Two triangle example image\",\"author\":\"MATHEUS NAVARRO\",\"organization\":\"Shutterstock\",\"url\":\"https:\/\/www.shutterstock.com\/image-vector\/geometric-problem-find-value-x-similarity-2149824621\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":71,"module-header":"learn_it","content_attributions":[{"type":"copyrighted_video","description":"Two triangle example image","author":"MATHEUS NAVARRO","organization":"Shutterstock","url":"https:\/\/www.shutterstock.com\/image-vector\/geometric-problem-find-value-x-similarity-2149824621","project":"","license":"arr","license_terms":""}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2995"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":45,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2995\/revisions"}],"predecessor-version":[{"id":15643,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2995\/revisions\/15643"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/71"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2995\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=2995"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=2995"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=2995"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=2995"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}